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Question: If the function f(x) = \(\frac{t + 3x - x^{2}}{x - 4}\), where t is a parameter that has a minimum a...

If the function f(x) = t+3x−x2x−4\frac{t + 3x - x^{2}}{x - 4}, where t is a parameter that has a minimum and maximum, then the range of value of t is

A

(0, 4)

B

(0, ¥)

C

(–¥, 4)

D

None

Answer

(–¥, 4)

Explanation

Solution

f(x) = t+3x−x2x−4\frac{t + 3x - x^{2}}{x - 4};

f ¢(x) = (x−4)(3−2x)−(t+3x−x2)(x−4)2\frac{(x - 4)(3 - 2x) - (t + 3x - x^{2})}{(x - 4)^{2}}

for max. or min. f ¢(x) = 0

–2x2 + 11x –12 –t –3x + x2 = 0

–x2 + 8x – (12 + t) = 0

for one maxima & minima

D > 064 – 4 (12 + t) > 0

16 – 12 – t > 0 ̃ 4 > t or t < 4